Hereafter, the random walk operator on graphs [1]}, [2]}, [3]} is proposed as being a suitable reference operator for extending the framework of signal processing on directed graphs. Like the graph Laplacian, the random walk operator is associated to the notion of diffusion. It is as well defined on directed graphs as on undirected ones. Assuming that the random walk operator is diagonalizable, it potentially admits conjugated complex eigenvectors as well as associated conjugated complex eigenvalues. Our framework is based on the fact that the variational analysis of the eigenvectors of the random walk operator through their associated Dirichlet energy [2]} is related to the real part of the eigenvalue [5]}. Therefore, the Dirichlet energy of a given eigenvector of the random walk operator on a directed graph can be considered as its frequency, using the same analogy as in [6]}, now for directed graphs. Provided with this notion of frequency, the eigenvectors of the random walk operator form a suitable Fourier basis on directed graphs. It is furthermore consistent with the undirected setting when the Fourier basis is based on the random-walk Laplacian operator (see section REF ).
